# Constraint-preserving hybrid finite element methods for Maxwell's equations

## TL;DR

This paper introduces a family of hybrid finite element methods for Maxwell's equations that strongly preserve physical constraints like charge conservation and magnetic monopole nonexistence, improving accuracy and reliability.

## Contribution

The authors develop a domain decomposition-based hybrid finite element framework that ensures strong preservation of Maxwell's physical constraints, including a hybridized Nédélec method with enhanced constraint enforcement.

## Key findings

- Methods strongly preserve divergence constraints in simulations.
- Numerical experiments confirm improved constraint preservation.
- Superconvergence observed with hybrid post-processing enhances magnetic field estimates.

## Abstract

Maxwell's equations describe the evolution of electromagnetic fields, together with constraints on the divergence of the magnetic and electric flux densities. These constraints correspond to fundamental physical laws: the nonexistence of magnetic monopoles and the conservation of charge, respectively. However, one or both of these constraints may be violated when one applies a finite element method to discretize in space. This is a well-known and longstanding problem in computational electromagnetics. We use domain decomposition to construct a family of primal hybrid finite element methods for Maxwell's equations, where the Lagrange multipliers are shown to correspond to a numerical trace of the magnetic field and a numerical flux of the electric flux density. Expressing the charge-conservation constraint in terms of this numerical flux, we show that both constraints are strongly preserved. As a special case, these methods include a hybridized version of N\'ed\'elec's method, implying that it preserves the constraints more strongly than previously recognized. These constraint-preserving properties are illustrated using numerical experiments in both the time domain and frequency domain. Additionally, we observe a superconvergence phenomenon, where hybrid post-processing yields an improved estimate of the magnetic field.